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57
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
Level Planar Embedding in Linear Time
, 1999
"... A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be ..."
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Cited by 20 (0 self)
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A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i, all v V i are drawn on the line li  {(x, k  i) ] x ), the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to
An O(m log n)Time Algorithm for the Maximal Planar Subgraph Problem
, 1993
"... Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C1 ..."
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Cited by 17 (0 self)
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Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [15], Wu defined the problem of planar graphs in terms of the following four subproblems: ################## 1 This work was partly supported by ThomsonCSF/DSE and by the National Science Foundation under grant CCR9002428. 2. Research at Princeton University partially supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, grant NSFSTC8809648, and the Office of Naval Research, contract N0001487K0467.    2  P1. Decide whether a connected graph G is planar. P2. Find a minimal set of edges the removal of which will render the remaining part of G planar. P3. Gi...
Drawing SeriesParallel Graphs on a Box
 The University of Lethbridge
, 1997
"... A box is a restricted portion of the threedimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A boxdrawing of a graph is a straightline crossingfree drawing where vertices are located at integer grid points along the four lines. It is known t ..."
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Cited by 17 (3 self)
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A box is a restricted portion of the threedimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A boxdrawing of a graph is a straightline crossingfree drawing where vertices are located at integer grid points along the four lines. It is known that some planar graphs with triconnected components do not admit a boxdrawing. This paper shows that even structurally simpler planar graphs, namely seriesparallel graphs, are not boxdrawable in general. On the positive side, it is proved that every seriesparallel graph whose vertices have maximum degree at most three is boxdrawable. A drawing algorithm is presented that computes a box drawing of a 3planar seriesparallel graph in optimal time and with optimal volume.
Optimal reduction of twoterminal directed acyclic graphs
 SIAM Journal on Computing
, 1992
"... Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O ..."
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Cited by 14 (1 self)
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Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2"5) algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NPhard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly seriesparallel.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Pitfalls of using PQTrees in Automatic Graph Drawing
, 1997
"... A number of erroneous attempts involving PQtrees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particula ..."
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Cited by 10 (0 self)
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A number of erroneous attempts involving PQtrees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particular, we examine erroneous usage of the PQtree data structure in algorithms for computing maximal planar subgraphs and an algorithm for testing leveled planarity of leveled directed acyclic graphs with several sources and sinks.
Fully Dynamic Planarity Testing in Planar Embedded Graphs
, 1993
"... We present the first data structure to maintain an embedded planar graph under arbitrary edge insertions, arbitrary edge deletions and queries that test whether the insertion of a new edge would violate the planarity of the embedding. Our data structure supports online updates and queries on an nv ..."
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Cited by 7 (1 self)
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We present the first data structure to maintain an embedded planar graph under arbitrary edge insertions, arbitrary edge deletions and queries that test whether the insertion of a new edge would violate the planarity of the embedding. Our data structure supports online updates and queries on an nvertex embedded planar graph in O(log 2 n) worstcase time, it can be built in O(n) time and requires O(n) space. This work was supported in part by ESPRIT BRA ALCOM II under contract no. 7141 and by the Italian MURST Project "Algoritmi, Modelli di Calcolo e Strutture Informative". A preliminary version of this paper was presented at the 1st European Symposium on Algorithms, Bad Honnef, Bonn, Germany [10]. y Dipartimento di Informatica e Sistemistica, Universit`a di Roma "La Sapienza", Roma, Italy. On leave from IBM T.J. Watson Research Center. z Department of Computer Science, Princeton University, Princeton, NJ 08544, USA. The research of this author was supported by a NATO Scienc...
Eager stOrdering
, 2002
"... Given a biconnected graph G = (V; E) and an edge fs; tg 2 E, an stordering is an ordering v1, ..., vn of V such that s = v1 , t = vn , and every other vertex has both a highernumbered and a lower numbered neighbor. Previous lineartime algorithms to compute such orderings are based on a preprocess ..."
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Cited by 7 (1 self)
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Given a biconnected graph G = (V; E) and an edge fs; tg 2 E, an stordering is an ordering v1, ..., vn of V such that s = v1 , t = vn , and every other vertex has both a highernumbered and a lower numbered neighbor. Previous lineartime algorithms to compute such orderings are based on a preprocessing step in which depthfirst search is used to compute lowpoints. The actual ordering is determined only in a second pass over the graph. We present
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 6 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �